Combinatorial local convexity implies convexity in finite dimensional CAT(0) cubed complexes

We give a proof to the following theorem, which is well-known among experts: A connected subcomplex $W$ of a finite dimensional CAT(0) cubed complex $X$ is convex if and only if Lk$(v,W)$ is a full subcomplex of Lk$(v,X)$ for every vertex $v$ of $W$.